Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54764 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,5)(6,9,10,8)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28)(29,34)(30,36)(31,33)(32,35)(37,41)(38,42)(39,40);(4,5)(8,9)(12,16)(13,19)(14,23)(24,26)(27,29,33,30)(28,34,31,36)(32,38,40,37)(35,42,39,41);(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(3,5,7,4)(6,9,10,8)(11,13,15,19)(12,17,16,20)(14,21,23,25)(18,24,22,26)(27,31,33,28)(29,36,30,34)(32,39,40,35)(37,42,38,41);(3,6)(4,8)(5,9)(7,10)(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,32,33,40)(28,35,31,39)(29,37,30,38)(34,41,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006984);